This project involves collaborative work with NIH scientists, in which sophisticated mathematical tools or an acquaintance with theoretical methods in the physical sciences are required for the research. Drs. George Weiss, Marian Bogunya, and Alexander Berezhkovskii (NIDDK) have worked on a variety of problems required to develop the theory of single-molecule spectroscopy. So far an exact solution has been found for the simplest possible two-state system which is an isomerization scheme with first-order transitions. Considerable effort has been spent in developing theoretical approaches to deal with more complicated chemical systems. This requires expansion of the theory of cumulative residence times in one or more states of a multistate system. To date we have published several papers on this subject. A second topic relates to the effects of surfaces on mathematical properties required to describe photon trajectories in tissue, which is relevant to the use of laser light for diagnostic purposes. Many mathematical details of the theory become quite complicated when there is a boundary separating tissue from an external environment. We are currently working on problems in this general area.